On strongly-non-reflexive groups

A strong negative answer is given to the old question of whether every dual group is reflexive. Using ◊_ω1 a groupA is constructed so thatA, A*, A**, andA*** are weakly ω₁-separable groups of cardinalityω ₁ andA* is not isomorphic toA***. Research partially supported by NSF Grant No. DMS-8400451. Research partially supported by NSERC Grant No. A8948.     

Hereditarily separable groups and monochromatic uniformization

We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality ?₁. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove that it is consistent that every Whitehead group is free but not every hereditarily separable group is free. We also show that the fact that ? is a p.i.d. with infinitely many primes is essential for this result.     

Uniformization and the diversity of Whitehead groups

Techniques of uniformization are used to prove that it is not consistent that the Whitehead groups of cardinality ℵ₁ are exactly the strongly ℵ₁-free groups. Some consequences of the assumption that every strongly ℵ₁-free group of cardinality ℵ₁ is Whitehead are derived. Other results about uniformization are also proved. Research partially supported by NSERC grant #9848. Research partially supported by the BSF. The authors thank Rutgers University for its support. Publication #441.     

Almost disjoint abelian groups

Under various set-theoretic hypotheses we construct families of maximal possible size of almost free abelian groups which are pairwise almost disjoint, i.e. there is no non-free subgroup embeddable in two of them. We show that quotient-equivalent groups cannot be almost disjoint, but we show how to construct maximal size families of quotient-equivalent groups of cardinality ℵ₁, which are mutually non-embeddable. Dedicated to the memory of Abraham Robinson on the tenth anniversary of this death First and third authors acknowledge assistance from the US-Israel Binational Science Foundation, Grant No. 1110. First author partially supported by NSF Grant No. MCS-8003691. Second author acknowledges support from the National Science and Engineering Research Council of Canada, Grant No. U0075     

Torsion Modules, Lattices and P-Points

Answering a long-standing question in the theory of torsionmodules, we show that weakly productively bounded domains arenecessarily productively bounded. (See the Introduction fordefinitions.) Moreover, we prove a twin result for the ideallattice L of a domain equating weak and strong global intersectionconditions for families (X_i)_iI of subsets of L with the propertythat _iI A_i 0 whenever A_iX_i. Finally, we show that for domainswith Krull dimension (and countably generated extensions thereof),these lattice-theoretic conditions are equivalent to productiveboundedness. 1991 Mathematics Subject Classification 03E05,06A23, 13C12, 16U20, 16P60.     

The Kaplansky test problems for -separable groups

We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for -separable abelian groups of cardinality . In fact, there is an -separable abelian group such that is isomorphic to but not to . We also derive some relevant information about the endomorphism ring of .

     

Baer modules over valuation domains

Summary A module B over a commutative domain R is said to be a Baer module if Ext _R ¹ (B, T)=0for all torsion R-modules T. The case in which R is an arbitrary valuation domain is investigated, and it is shown that in this case Baer modules are necessarily free. The method employed is totally different from Griffith's method for R=Z which breaks down for non-hereditary rings.This research was partially supported by NSF Grants DMS-8400451 and DMS-8500933.     

Algebraic closure operators and strong amalgamation bases

Axioms for a closure operator are given and it is proved that, for suitable classes, a unique largest closure operator exists. Necessary and sufficient conditions in terms of closure properties are given for structures to be strong amalgamation bases. Research supported by NSF GP 34091X Presented by B. Jónsson     

On singular compactness

We present some applications of Shelah's singular compactness theorem to algebraic situations where the Shreier property fails. The principal application is to valuated vector spaces, where we make use of an alternate, unpublished, version of Shelah's theorem. Research partially supported by NSF Grant No. MCS 80-03591 and by the US-Israel BSF. Presented by L. Fuchs.